Spatially localized noise adaptive smoothing of emission tomography images

ABSTRACT

A method and system for spatially localized, noise-adaptive smoothing of filtered back-projection (FBP) nuclear images includes smoothing a FBP image on a pixel-by-pixel basis using a shift-invariant kernel such as a Gaussian kernel. The width of the kernel may be varied on a pixel basis, such that the smoothed image contains a constant SNR over the entire image.

TECHNICAL FIELD

The current invention is in the field of medical imaging. Particularly,the invention relates to techniques for image reconstruction fromemission projection data and reduction of statistical fluctuations inthe resulting reconstructed tomographic images.

BACKGROUND OF THE INVENTION

Medical imaging is one of the most useful diagnostic tools available inmodern medicine. Medical imaging allows medical personnel tonon-intrusively look into a living body in order to detect and assessmany types of injuries, diseases, conditions, etc. Medical imagingallows doctors and technicians to more easily and correctly make adiagnosis, decide on a treatment, prescribe medication, perform surgeryor other treatments, etc.

There are medical imaging processes of many types and for many differentpurposes, situations, or uses. They commonly share the ability to createan image of a bodily region of a patient, and can do so non-invasively.Examples of some common medical imaging types are nuclear medical (NM)imaging such as positron emission tomography (PET) and single photonemission computed tomography (SPECT). Using these or other imaging typesand associated machines, an image or series of images may be captured.Other devices may then be used to process the image in some fashion.Finally, a doctor or technician may read the image in order to provide adiagnosis.

A PET camera works by detecting pairs of gamma ray photons in timecoincidence. The two photons arise from the annihilation of a positronand electron in the patient's body. The positrons are emitted from aradioactive isotope that has been used to label a biologically importantmolecule like glucose (a radiopharmaceutical). Hundreds of millions suchdecays occur per second in a typical clinical scan. Because the twophotons arising from each annihilation travel in opposite directions,the rate of detection of such coincident pairs is proportional to theamount of emission activity, and hence glucose, along the lineconnecting the two detectors. In a PET camera the detectors aretypically arranged in rings around the patient. By consideringcoincidences between all appropriate pairs of these detectors, a set ofprojection views can be formed each element of which represents a lineintegral, or sum, of the emission activity in the patient's body along awell defined path. These projections are typically organized into a datastructure called a sinogram, which contains a set of plane parallelprojections at uniform angular intervals around the patient. A threedimensional image of the radiophamaceutical's distribution in the bodycan then be reconstructed from these data.

A SPECT camera functions similarly to a PET camera in many ways, butdetects only single photons rather than coincident pairs. For thisreason, a SPECT camera must use a lead collimator with holes, placed infront of its detector panel, to define the lines of response in itsprojection views. One or more such detector panel/collimatorcombinations rotates around a patient, creating a series of planarprojections each element of which represents a sum of the emissionactivity, and hence biological tracer, along the line of responsedefined by the collimation. As with PET, these data can be organizedinto sinograms and reconstructed to form an image of theradiopharmaceutical tracer distribution in the body.

The aim of the reconstruction process is to retrieve the spatialdistribution of the radiopharmaceutical from the projection data. Themain reconstruction step involves a process known as back-projection. Insimple back-projection, an individual data sample is back-projected bysetting all the image pixels along the line of response pointing to thesample to the same value. In less technical terms, a back-projection isformed by smearing each view back through the image in the direction itwas originally acquired. The back-projected image is then taken as thesum of all the back-projected views. Regions where back-projection linesfrom different angles intersect represent areas which contain a higherconcentration of radiopharmaceutical.

While back-projection is conceptually simple, it does not by itselfcorrectly solve the reconstruction problem. A simple back-projectedimage is very blurry; a single point in the true image is reconstructedas a circular region that decreases in intensity away from the center.In more formal terms, the point spread function of back-projection iscircularly symmetric, and decreases as the reciprocal of its radius.

Filtered back-projection (FBP) is a technique to correct the blurringencountered in simple back-projection. Each projection view is filteredbefore the back-projection step to counteract the blurring point spreadfunction. That is, each of the one-dimensional views is convolved with aone-dimensional filter kernel (e.g. a “ramp” filter) to create a set offiltered views. These filtered views are then back-projected to providethe reconstructed image, a close approximation to the “correct” image.

The inherent randomness of radioactive decay and other processesinvolved in generating nuclear medical image data results in unavoidablestatistical fluctuations (noise) in PET or SPECT data. This is afundamental problem in clinical imaging that is dealt with through someform of smoothing of the data. In FBP this is usually accomplished bymodifying the filter kernel used in the filtering step by applying alow-pass windowing function to it. This results in a spatially uniform,shift-invariant smoothing of the image that reduces noise, but may alsodegrade the spatial resolution of the image. A disadvantage of thisapproach is that the amount of smoothing is the same everywhere in theimage although the noise is not. Certain regions, e.g. where activityand detected counts are higher, may have relatively less noise and thusrequire less smoothing than others. Standard windowed FBP cannot adaptto this aspect of the data.

There are several alternatives to FBP for reconstructing nuclear medicaldata. In fact, most clinical reconstruction of PET images is now basedon some variant of regularized maximum likelihood (RML) estimationbecause of the remarkable effectiveness of such algorithms in reducingimage noise compared to FBP. In a sense, RML's effectiveness stems fromits ability to produce a statistically weighted localized smoothing ofan image. These algorithms have some drawbacks however: they arerelatively expensive because they must be computed iteratively; theygenerally result in poorly characterized, noise dependent, image bias,particularly when regularized by premature stopping (unconverged); andthe statistical properties of their image noise are difficult todetermine.

There remains a need in the art for improvement in image reconstructiontechniques to produce an image with statistically adaptive noisereduction similar to RML, but faster, with less computational complexityand better quantitative accuracy, similar to FBP.

SUMMARY OF THE INVENTION

A new statistically adaptive smoothing algorithm for FBP is providedthat equalizes the local relative noise level, or signal-to-noise ratio(SNR), across an image on a pixel-by-pixel basis. This new process isreferred to herein as SNR-equalized FBP (SNR-FBP). The localizedsmoothing produces images with visual qualities much more similar to RMLthan does conventional windowed FBP. In contrast to RML however, SNR-FBPhas well characterized spatial resolution and noise variance at everypoint in the image at all noise levels. Thus, while SNR-FBP may not havethe optimal statistical properties of a true maximum likelihood ormaximum a posteriori reconstruction, it may prove valuable fortime-constrained reconstructions where accurate quantitation with knownprecision and resolution is important.

The method for tomographic image reconstruction provided is based on twosimple principles. First, the raw data in nuclear medicalimaging—detected count rates—are Poisson distributed random variableswhose (noise) variance can be accurately estimated. (For a Poissonvariable, the variance is equal to the mean.) Second, for a linearreconstruction algorithm, such as FBP, any reconstructed image elementcan be ultimately represented as a linear combination of the raw data.Therefore, the noise variance in the image element can be computed froma simple propagation of errors from the raw data. This variance can becontrolled by adjusting the width of the smoothing kernel used in thereconstruction. As the width increases, the variance decreases. Byreconstructing the data using several different smoothing filters andinterpolating between the resulting mean and variance imagesindependently in each pixel, a specified relative noise level. or SNR,can be achieved in each image element.

Let R_(α) ^(ij) represent the reconstructed mean image value in a pixelindexed by (i,j), and let V_(α) ^(ij) represent the computed noisevariance in that pixel element. α is a parameter indicating the width ofthe smoothing kernel employed in the reconstruction. The method includesreconstructing R_(α) ^(ij) and V_(α) ^(ij) for several different valuesof α. The method further includes the steps of forming a signal-to-noiseratio (SNR), SNR_(α) ^(ij)=R_(α) ^(ij)/√{square root over (V_(α)^(ij))}, for each R_(α) ^(ij) and V_(α) ^(ij), selecting a target valuefor the SNR of the output image, and interpolating between the R_(α)^(ij) values whose corresponding SNR_(α) ^(ij) values bracket the targetSNR using a model of the variation of R_(α) ^(ij) with SNR_(α) ^(ij).

A system employing the method of the current invention is also provided.The system includes a nuclear medical imaging device, a processor forreceiving data from the image scanner, and software running on theprocessor. The software is capable of performing the steps of the methodof the current invention.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will now be described in greater detail in the followingby way of example only and with reference to the attached drawings, inwhich:

FIG. 1 is a schematic of the method of the current invention.

FIGS. 2 and 3 depict PET image slices from two patients using differentreconstruction and filtering techniques.

FIG. 4 depicts TOF-PET image slices reconstructed using eitherconventional filtering or SNR equalization.

FIG. 5 depicts image slices of the FWHMs used for the SNR-FBP images ofFIG. 4.

FIG. 6 is a diagram of a system according to a possible embodiment ofthe current invention.

DETAILED DESCRIPTION OF THE INVENTION

As required, disclosures herein provide detailed embodiments of thepresent invention; however, the disclosed embodiments are merelyexemplary of the invention that may be embodied in various andalternative forms. Therefore, there is no intent that specificstructural and functional details should be limiting, but rather theintention is that they provide a basis for the claims and as arepresentative basis for teaching one skilled in the art to variouslyemploy the present invention.

FIG. 1 is a flow chart of a method 100 according to the presentinvention. At step 110, nuclear medical image projection data isobtained. The projection data may be from a PET scanner or a SPECTscanner. The PET data may optionally include time-of-flight (TOF)information. At step 120 the projection data are processed using astandard linear reconstruction algorithm to produce an image of theestimated mean value of the radiopharmaceutical concentration, R_(α)^(ij), or related quantity. For SPECT or non-TOF PET, the algorithm maybe FBP. For TOF PET, the algorithm may be confidence-weighted filteredback-projection or another similar algorithm known to those skilled inthe art. In either case, the reconstruction process incorporates ashift-invariant smoothing kernel of variable width, α. A typicalfunction form for such a kernel would be a Gaussian, but many otherforms are possible. The reconstruction is repeated for several differentvalues of α. Also at step 120 the noise variance image, V_(α) ^(ij),corresponding to each smoothing kernel width is computed.

The mean image value at pixel (i, j), R_(α) ^(ij), depends on the width,α, of the kernel. For FBP this image value can be expressed in terms ofthe measured projection data according to:

${R_{\alpha}^{ij} = {\sum\limits_{lm}{{h_{\alpha}\left( {l - {i\; \cos \; \theta_{m}} + {j\; \sin \; \theta_{m}}} \right)}a_{lm}{n_{lm}\left( {p_{lm} - d_{lm} - s_{lm}} \right)}}}},$

where l and m are respectively the radial and angular sinogram binindices, θ_(m) is the projection angle, a and n are respectively theattenuation correction and normalization factors, and p, d and s are theprompt, randoms, and scattered coincidence data respectively. h_(α)(l)is the discrete filtering kernel given by the convolution:

${{h_{\alpha}(l)} = {\sum\limits_{k}{C_{k}^{l}{g_{\alpha}(k)}}}},$

where C_(k) ^(l) is the standard FBP kernel corresponding to a rampfilter, k is a radial bin index, and g_(α) is the smoothing kernel. Fora Gaussian smoothing kernel:

${{g_{\alpha}(k)} = {\frac{1}{\sqrt{2\; \pi}\sigma}{\exp \left( {- \frac{k^{2}}{2\; \sigma^{2}}} \right)}}},$

whose spatial full width at half maximum (FWHM) is α=√{square root over(2σ²(4 ln 2))}.

Assuming that the prompt data are uncorrelated and Poisson distributed,and that a noise-free estimate of the scatter rate is available, animage of the variance of R_(α) ^(ij), V_(α) ^(ij), may be computedaccording to:

$V_{\alpha}^{ij} = {\sum\limits_{lm}{{h_{\alpha}^{2}\left( {l - {i\; \cos \; \theta_{m}} + {j\; \sin \; \theta_{m}}} \right)}a_{lm}^{2}{{n_{lm}^{2}\left( {p_{lm} + \left\lbrack d_{lm} \right\rbrack} \right)}.}}}$

The term d_(lm) may be required for delayed coincidence subtraction, butmay be deleted if a noise free estimate of the randoms is used. V_(α)^(ij) is not the same as a pixel-to-pixel variance over R_(α) ^(ij) inthe original image, but may be an estimate of the ensemble variance ofthe value of R_(α) ^(ij).

Given the mean and variance images, an SNR image may be formed fromtheir ratio:

${SNR}_{\alpha}^{ij} = {\frac{R_{\alpha}^{ij}}{\sqrt{V_{\alpha}^{ij}}}.}$

For a given (fixed) kernel width, SNR_(α) ^(ij) may be higher in thoseregions of the image with higher emission activity, such as hot spots.For a given pixel, SNR_(α) ^(ij) tends to increase monotonically as thewidth α of the kernel increases. If a particular value of SNR isspecified, e.g. SNR=5, it may be possible to achieve this value in anypixel by appropriately adjusting α. An image may then be formed, eachpixel of which is drawn from the particular R_(α) ^(ij) that achievesthe specified SNR at that point.

There then may be an SNR-equalized image with constant SNR but varyingα, (α_(ij)). The smoothing kernel needed to achieve this SNR-FBP imagemay be narrower in higher intensity regions and broader in lowintensity, noisier regions resulting in statistically adaptivesmoothing. Because SNR_(α) tends to vary smoothly with α, the SNR-FBPimage can be computed by reconstructing R_(α) ^(ij) and V_(α) ^(ij) at afew values of α (step 120), forming an SNR image SNR_(α) ^(ij) for eachR_(α) ^(ij) and V_(α) ^(ij) (step 130), selecting a target SNR value forthe output image (step 140), and interpolating between the calculatedR_(α) ^(ij) a image values at step 150.

Each pixel may be interpolated between the R_(α) ^(ij) values whoseSNR_(α) ^(ij) corresponding a values bracket the target SNR using amodel, which may be linear, of the variation of R_(α) ^(ij) with SNR_(α)^(ij). The kernel widths can also be interpolated to give the effectiveFWHM resolution at each point, α_(ij). The overall smoothness level inthe image is controlled by the SNR value, with higher values givingsmoother images. In a first embodiment, using the interpolated values, aSNR-equalized image R_(eq) ^(ij) may be formed.

It is possible that the target SNR may not be achievable at some pointwithin the image for any reasonable value of α_(ij). To deal with suchcases, the permitted values of α_(ij) may be constrained to lie withincertain limits, and the value giving an SNR_(α) ^(ij) closest to thedesired target value used, thus determining R_(α) ^(ij) at that point.

The technique is not limited to conventional FBP but could be applied toany reconstruction algorithm for which the R_(α) ^(ij) and V_(α) ^(ij)images can be readily computed. This capability has been demonstratedfor linear time-of-flight (TOF) reconstruction of PET, and thus themethodology for producing an SNR-equalized image can be equally wellapplied in this case.

In a second embodiment, an image based on any metric that is a functionof R_(α) ^(ij) and V_(α) ^(ij) may be formed. For example, the selectedvalue of SNR may be treated as a threshold. The image may be smoothedusing a default value of α, but for those pixels for which SNR_(α) ^(ij)was less than the desired minimum value, α_(ij) would be increased untilthe threshold SNR value was met.

As a second example, α_(ij) could be chosen at each point so thatSNR_(α) ^(ij) was increased by a constant factor. Statistical functionsother than SNR may also be used.

In a third embodiment, the statistical information provided by themethod may be examined directly. For instance the variance or SNR imagescan be examined at different resolutions that are computed during thegeneration of the SNR-FBP image.

FIGS. 2 and 3 show results from two image slices of a PET patient study.The reconstruction was non-TOF FBP. In each figure, the top row ofimages 210, 220, 230, and 310, 320, and 330 show standard FBPpost-smoothed with Gaussian kernels of different widths, illustratingtypical conventional clinical image processing. The images on the bottomleft 240 and 340 are the SNR-equalized FBP image, with SNR=5. Forcomparison, a standard clinical RML iteratively reconstructed image 250,350 is shown on the lower right.

FIG. 4 is an example of SNR equalization in linear, confidence-weightedTOF-PET reconstruction. The images in the top row 410 and 420 areconventional TOF reconstruction with post-smoothing by a Gaussian kernelof 8 or 16 mm respectively. The bottom row shows SNR-equalizedreconstructions at two values of target SNR, SNR=6 (430) and SNR=9(440).

FIG. 5 shows images of the FWHMs α_(ij) used to produce theSNR-equalized images of FIG. 4. The range used is from 4 to 60 mm with60 mm corresponding to white.

FIG. 6 is a diagram of a system 600 using the method 100 of the presentinvention. The system includes a medical imaging device 610, i.e. a PETscanner or a SPECT scanner. The medical imaging device 610 may beconnected to a processor 620 that receives the projection data from themedical imaging device. The processor 620 may have software running onit that is capable of executing the method 100 of the present invention.The software processes the projection data from the medical imagingdevice 610 and outputs an adaptively smoothed, SNR-equalized image.

The invention having been thus described, it will be apparent to thoseskilled in the art that the same may be varied in many ways withoutdeparting from the spirit and scope of the invention. Any and all suchvariations are intended to be included within the scope of the followingclaims.

1. A method for spatially localized noise-adaptive smoothing of nuclearmedicine tomographic images, comprising: obtaining a mean value R_(α)^(ij) for each pixel (i,j) of a reconstructed image by reconstructingthe corresponding nuclear medicine data using a shift-invariantsmoothing kernel of width α; obtaining a variance V_(α) ^(ij) of eachR_(α) ^(ij); and forming a signal-to-noise ratio image SNR_(α) ^(ij)from the ratio${SNR}_{\alpha}^{ij} = {\frac{R_{\alpha}^{ij}}{\sqrt{V_{\alpha}^{ij}}}.}$2. The method of claim 1, further comprising (a) reconstructing allR_(α) ^(ij) and V_(α) ^(ij) for two or more different values of α withinan interval α_(min)≦α≦α_(max); (b) forming an SNR image, SNR_(α) ^(ij),for each R_(α) ^(ij) and V_(α) ^(ij); (c) selecting a target value forthe SNR of the output image; and (d) determining a value of R_(α) ^(ij)corresponding to the target value of SNR by interpolating between theR_(α) ^(ij) values whose corresponding SNR_(α) ^(ij) values bracket thetarget SNR using a model of the variation of R_(α) ^(ij) with SNR_(α)^(ij). In the case that the target value of SNR is not bracketed bycomputed SNR_(α) ^(ij) values, the R_(α) ^(ij) value corresponding tothe target SNR is the one whose corresponding value of SNR_(α) ^(ij) isclosest to the target SNR value.
 3. The method of claim 2, wherein saidstep (a) reconstructing R_(α) ^(ij) and V_(α) ^(ij) for two or moredifferent values of α further comprises:${R_{\alpha}^{ij} = {\sum\limits_{lm}{{h_{\alpha}\left( {l - {i\; \cos \; \theta_{m}} + {j\; \sin \; \theta_{m}}} \right)}a_{lm}{n_{lm}\left( {p_{lm} - d_{lm} - s_{lm}} \right)}}}},$wherein: l and m are respectively the radial and angular sinogram binindices; θ_(m) is the projection angle; a and n are respectively theattenuation correction and normalization factors; p, d and s arerespectively the prompt, randoms, and scattered coincidence data;h_(α)(l) is the discrete filtering kernel wherein:${{h_{\alpha}(l)} = {\sum\limits_{k}{C_{k}^{l}{g_{\alpha}(k)}}}},$wherein: C_(k) ^(l) is the standard filtered back-projection kernelcorresponding to a ramp filter; k is a radial bin index; and g_(α)(k) isa smoothing kernel of width α; and${V_{\alpha}^{ij} = {\sum\limits_{lm}{{h_{\alpha}^{2}\left( {l - {i\; \cos \; \theta_{m}} + {j\; \sin \; \theta_{m}}} \right)}a_{lm}^{2}{n_{lm}^{2}\left( {p_{lm} + {\gamma \; d_{lm}}} \right)}}}},$wherein γ is a number between 0 and 1, with γ=0 for noise-free randomscorrection and γ=1 for delayed coincidence subtraction.
 4. The method ofclaim 3 wherein said smoothing kernel g_(α)(k) is a Gaussian of width α,${g_{\alpha}(k)} = {\frac{1}{\sqrt{2\; \pi}\sigma}{\exp\left( {- \frac{k^{2}}{2\; \sigma^{2}}} \right)}}$whose spatial full width at half maximum isα=√{square root over (2σ²(4 ln 2))}.
 5. The method of claim 2, furthercomprising: (e) using the interpolated values of R_(α) ^(ij) to form anSNR-equalized image, R_(eq) ^(ij), wherein SNR_(α) ^(ij) isapproximately the same for an appreciable set of pixels.
 6. The methodof claim 1, wherein the tomographic image is a PET image.
 7. The methodof claim 2, wherein the model of step (d) is a linear model.
 8. Themethod of claim 2, wherein step (d) further comprises interpolatingkernel widths to give the effective FWHM resolution at each point,α_(ij).
 9. The method of claim 2, further comprising: (e) using theselected value for the SNR as a threshold; (f) reconstructing R_(α)^(ij), V_(α) ^(ij) and SNR_(α) ^(ij) images using a particular defaultvalue of α; (g) selecting the set of pixels in these images for whichthe value of SNR_(α) ^(ij) is less than the target SNR value; and (h)substituting the said value of R_(α) ^(ij) corresponding to the targetvalue of SNR for each pixel in said set.
 10. The method of claim 9,wherein the tomographic image is a PET image.
 11. The method of claim 9,wherein the model of step (d) is a linear model.
 12. The method of claim9, wherein step (d) further comprises interpolating kernel widths togive the effective FWHM resolution at each point, α_(ij).
 13. A systemfor producing spatially localized noise-adaptive smoothing of nuclearmedicine tomographic images, comprising: a nuclear medicine tomographicimaging device; a processor for receiving data from the image device;and software executing on the processor, which obtains a mean valueR_(α) ^(ij) for each pixel (i,j) of a reconstructed image byreconstructing the corresponding nuclear medicine data using ashift-invariant smoothing kernel of width a; obtains a variance V_(α)^(ij) of each R_(α) ^(ij); and forms a signal-to-noise ratio imageSNR_(α) ^(ij) from the ratio${SNR}_{\alpha}^{ij} = {\frac{R_{\alpha}^{ij}}{\sqrt{V_{\alpha}^{ij}}}.}$14. The system of claim 13, wherein said software reconstructs R_(α)^(ij) and V_(α) ^(ij) for two or more different values of αwithin aninterval α_(min)≦α≦α_(max), forms a signal-to-noise ratio (SNR) image,SNR_(α) ^(ij), for each R_(α) ^(ij) and V_(α) ^(ij), selects a targetvalue for the SNR of the output image, and determines a value of R_(α)^(ij) corresponding to the target value of SNR by interpolating betweenthe R_(α) ^(ij) values whose corresponding SNR_(α) ^(ij) values bracketthe target SNR using a model of the variation of R_(α) ^(ij) withSNR_(α) ^(ij). In the case that the target value of SNR is not bracketedby computed SNR_(α) ^(ij) values, the R_(α) ^(ij) value corresponding tothe target SNR is the one whose corresponding value of SNR_(α) ^(ij) isclosest to the target SNR value.
 15. The system of claim 14, wherein:${R_{\alpha}^{ij} = {\sum\limits_{lm}{{h_{\alpha}\left( {l - {i\; \cos \; \theta_{m}} + {j\; \sin \; \theta_{m}}} \right)}a_{lm}{n_{lm}\left( {p_{lm} - d_{lm} - s_{lm}} \right)}}}},$wherein: l and m are respectively the radial and angular sinogram binindices; θ_(m) is the projection angle; a and n are respectively theattenuation correction and normalization factors; p, d and s arerespectively the prompt, randoms, and scattered coincidence data;h_(α)(l) is the discrete filtering kernel wherein:${{h_{\alpha}(l)} = {\sum\limits_{k}{C_{k}^{l}{g_{\alpha}(k)}}}},$wherein: C_(k) ^(l) is the standard filtered back-projection kernelcorresponding to a ramp filter; k is a radial bin index; and g_(α)(k) isa smoothing kernel of width a; and${V_{\alpha}^{ij} = {\sum\limits_{lm}{{h_{\alpha}^{2}\left( {l - {i\; \cos \; \theta_{m}} + {j\; \sin \; \theta_{m}}} \right)}a_{lm}^{2}{n_{lm}^{2}\left( {p_{lm} + {\gamma \; d_{lm}}} \right)}}}},$wherein γ is a number between 0 and 1, with γ=0 for noise-free randomscorrection and γ=1 for delayed coincidence subtraction.
 16. The systemof claim 15 wherein said smoothing kernel g_(α)(k) is a Gaussian ofwidth α,${g_{\alpha}(k)} = {\frac{1}{\sqrt{2\; \pi}\sigma}{\exp \left( {- \frac{k^{2}}{2\; \sigma^{2}}} \right)}}$whose spatial full width at half maximum isα=√{square root over (2σ²(4 ln 2))}.
 17. The system of claim 14, whereinthe software uses the interpolated values of R_(α) ^(ij) to form anSNR-equalized image, R_(α) ^(ij) wherein SNR_(α) ^(ij) is approximatelythe same for an appreciable set of pixels.
 18. The system of claim 13,wherein the imaging device is a PET scanner.
 19. The system of claim 14,wherein the interpolation model used by the software is a linear model.20. The system of claim 14, wherein the software interpolates kernelwidths to give the effective FWHM resolution at each point, α_(ij). 21.The system of claim 14, wherein the software uses the selected value forthe SNR as a threshold; reconstructs R_(α) ^(ij), V_(α) ^(ij) andSNR_(α) ^(ij) images using a particular default value of α; selects theset of pixels in these images for which the value of SNR_(α) ^(ij) isless than the target SNR value; and substitutes the said value of R_(α)^(ij) corresponding to the target value of SNR for each pixel in saidset.
 22. The system of claim 21, wherein the software interpolateskernel widths to give the effective FWHM resolution at each point,α_(ij).